Cooperation norms in multiple-stage punishment

Learning and Peer Effects

Andreas Nicklisch Irenaeus Wolff

Research Paper Series

Thurgau Institute of Economics and Department of Economics at the University of Konstanz

No. 54 june 2010

Cooperation norms in multiple-stage punishment

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-121843

URL: http://kops.ub.uni-konstanz.de/volltexte/2010/12184/

Cooperation norms in multiple-stage punishment ^∗

Andreas Nicklisch

Max Planck Institute for Research on Collective Goods, Bonn nicklisch@coll.mpg.de

and Irenaeus Wol

University of Konstanz, TWI wol@twi-kreuzlingen.ch

July 5, 2010

Abstract

We analyze the interplay between cooperation norms and people's punishment behavior in a social-dilemma game with multiple pun- ishment stages. By combining multiple punishment stages with self- contained episodes of interaction, we are able to disentangle the eects of retaliation and norm-related punishment. An additional treatment provides information on the norms bystanders use in judging punish- ment actions. Partly conrming previous ndings, punishment behav- ior and bystanders' opinions are guided by an absolute norm. This norm is consistent over decisions and punishment stages and requires full contributions. In the rst punishment stage, our results suggest a higher personal involvement of punishers, leading to a non-linearity dened by the punishers' contribution. In later punishment stages, the personal-involvement eect vanishes and retaliation kicks in. By- standers generally apply the same criteria in all stages, also favoring retaliation in response to harsh punishment actions.

Keywords: Experiment, public-good, punishment, social norms, volun- tary cooperation

JEL-Classication: C92, D63, H41

∗We are deeply indebted to Sophie Bade, Katharine Bendrick, Christoph Engel, Michael Kurschilgen, Bettina Rockenbach, Marie-Claire Villeval, and two anonymous referees for reading an earlier version of the paper and providing us with useful and detailed feedback.

We would further like to thank the participants of the IMEBE workshop 2008 in Alicante

1 Introduction

Norms (i.e., common understandings about obligatory, permitted, or forbid- den behavior)¹ inuence our behavior in many real-world scenarios. People entering buildings keep doors open for others, parents' nancial support for kindergarten initiatives is typically proportional to income as we expect the tax burden to be and men take their hats o when entering churches.

There are numerous other examples of how norms guide behavior in groups, so that economics has devoted a substantial amount of eort to analyzing the inuence of social norms in the last decades (important contributions include, e.g., Sugden, 1986, Sethi, 1996, or Sober & Wilson, 1998).

Of particular interest for the economist's study of norms is their interplay with individual incentives. The archetype of a potential conict between social norms and individual incentives is the social dilemma, where individual and collective interests are misaligned. Norm violations and others' responses to such violations have long been debated in the experimental literature in the context of decentralized sanctioning mechanisms. In this context, a norm is the (implicitly agreed upon) reference value of the cooperation level such that deviating from this cooperation target leads to the deviating players being sanctioned.²

Sanctions have been shown to foster and maintain voluntary cooperation in social dilemmas (seminal work has been provided by Ostrom et al., 1992, for common-pool resources, and Yamagishi, 1986, or Fehr & Gächter, 2000, for public goods). Our paper sets out to analyze explicitly the norms of cooperation prevailing in situations of this kind, and systematically compares potential norm candidates in an experiment tailored for this purpose. More precisely, we elicit the norms employed in sanctioning uncooperative behavior when there are multiple sanctioning stages, and examine whether other group members who are not directly involved in the punishment actions share the same norms for sanctioning.³

When thinking about cooperation norms in social-dilemma situations, one important distinction is that between relative and absolute norms. Rel- ative norms are variable reference points that rise and drop with the level of cooperation within the group. In contrast, absolute norms provide reference

1Cf. Ostrom (2000).

2Cf. the use of the term, e.g., by Carpenter and Matthews (2009).

3Note that we do not analyze how punished players react to sanctions that are justied according to the dierent norms. Evaluating reactions in this sense would be an interesting exercise, but would require that we assume the crucial norm in advance. Other authors have explored this interesting issue (e.g., Cinyabuguma et al., 2006, Ones & Putterman, 2007) which would go beyond the scope of our experimental design.

points for behavior independent of the group's current level of cooperation (for instance, there could be a norm always to cooperate fully). A relative- norm model would merely predict punishment to be observed until behavior has converged; an absolute-norm model also species the point of conver- gence.

Relative norms have been estimated in a number of studies, as theoretic models of prosocial behavior like the Fehr and Schmidt (1999) model suggest reference points to be relative in the above sense. This idea has received empirical support by studies such as Dawes et. al (2007) or Johnson et al. (2009) who nd evidence for egalitarian motives as a driving factor in costly punishment. In terms of norm choice, several authors rely on the average degree of cooperation within the group as the norm (Fehr & Gächter, 2000, 2002, Anderson & Putterman, 2006, and Sefton et al., 2007), while more recent studies focus on the degree of cooperation of the player who punishes (Herrmann et al., 2008, Egas & Riedl, 2008, Sutter et al., 2010, or Reuben & Riedl, 2009). Yet, little is known with respect to absolute norms and with respect to the question of whether relative or absolute norms guide cooperation and sanctions. An exception is Carpenter and Matthews (2009) who compare the predictive power of relative and absolute norms in explaining the sanctioning behavior. They show that by and large, absolute norms t the data better than relative norms. This nding, if robust, would challenge theoretical attempts to explain punishment behavior by existing models of pro-social behavior.

We extend the work of Carpenter and Matthews with respect to several important aspects. First, we are able to disentangle punishment related to a cooperative norm from acts of retaliation by (i) employing multiple sanc- tioning stages in conjunction with (ii) self-contained episodes of interaction (players change their interaction partners after each encounter). These fea- tures allow us to restrict counterpunishment actions to the individual episode of interaction, so that it does not directly aect the data obtained from later interactions. An interesting question following directly from the above is whether a persisting cooperation norm will play a role in higher iterations of punishment. Everyday experience tells us that the majority of situa- tions share the feature of iterative punishment being possible. Experimental research has shown that behavior in such sequences can dier substantially from the behavior typically observed in simple settings of a single sanctioning stage (e.g., Denant-Boemont et al., 2007, Nikiforakis, 2008, and Nikiforakis

& Engelmann, 2009).

The use of multiple sanctioning stages has a further advantage. It has long been known that a non-negligible fraction of punishment actions in social-

rized as antisocial punishment (e.g., Herrmann et al, 2008).⁴ Cinyabuguma et al. (2006) present some evidence that most antisocial punishment seems to stem from a sort of blind revenge. Thanks to our design, we are able to draw an even clearer picture and provide evidence on the social acceptabil- ity of retaliation. At the same time, we can largely rule out random errors as another possible source of high-contributor punishment suggested in the literature (cf. Fehr & Gächter, 2000).

On a second dimension, Carpenter and Matthews provide evidence that subjects employ dierent norms for the decisions of (i) whether to punish a player or not, and (ii) how hard they want to punish that particular player.

We further explore this eect by explicitly disentangling both decisions: in our setting, players rst announce to punish a certain player (at a cost), before deciding on the level of punishment in a second step. ⁵ Explicitly disentangling the decisions of whether to punish a player and by how much will be interesting, since it allows us to analyze the degree of consistency between the norms.

Finally, we provide additional insights on cooperation norms prevailing within groups by introducing an important treatment variation. In the stan- dard setting, norms are revealed only indirectly by those players actively sanctioning others. However, there are a substantial number of players who abstain from punishment actions. Still, it is not clear whether this abstention is owed to the players' norms of cooperation not being violated, or whether it is due to other reasons, such as an aversion to forcing others by means of punishment, or that the costs of punishment are higher than the player's disu- tility from the norm violation. As far as these players' cooperation norm is concerned, the traditional setting provides little evidence. In order to elicit a cooperation norm using data from all players, we introduce a treatment con- dition in which, for each punishment action announced, those group members who are neither the punisher nor the punishee with respect to that specic action have to voice their (dis-)agreement with it. In order not to render the announced (dis-)approvals of players completely arbitrary, but to create some commitment with respect to these statements on norm-related behavior, all players are informed about them. As such, agreements and disagreements have no formal consequences, while they provide additional information on norms within a group. Further details concerning the experimental design are discussed in the following two sections.

4Others call this form of punishment perverse, e.g., Cinyabuguma et al. (2006).

5Similarly, Masclet et al. (2009) employ a two-step procedure for punishment; in their case, punishment actions are publicly announced before the cooperation stage for each possible cooperation level. Subsequently, the announcer can revise her schedule in the actual punishment stage.

Our results indicate that in line with the ndings of Carpenter and Matthews, an absolute norm seems to organize the decisions relating to norm violations very well. Particularly, we observe an absolute norm dened by subjects' endowments that is consistent over dierent decisions and dierent actors. Often, a player's own contribution relative to the punished-to-be's contribution acts as an additional trigger in the rst iteration. However, this phases out quickly, as do contributions as a determinant of punishment- related decisions in general, but at a slower pace. In our treatment variation, bystanders' opinions rather than contribution dierences serve as the main determinant of the punishment level. However, opinions follow patterns that are remarkably similar to those found in punishers' announcements, which do not exhibit signicant dierences between treatments. Due to this fact, the observed behavior in both treatments is hardly distinguishable.

We observe punishment of high-contributors by lower-contributors pre- dominantly as a response to prior sanctioning by the former. This suggests that the perverse punishment observed in earlier studies is a form of `blind revenge' or `pre-emptive counterpunishment' rather than spiteful or com- petitive behavior or the consequence of a taste for conformity; only in our treatment variation, there are instances of perverse punishment. However, additional research is needed to clearly determine the reasons for this sur- prising treatment dierence.

The remaining article is organized as follows: section 2 introduces the game and presents our research questions. Section 3 describes the experi- mental design. Section 4 reports the results, while section 5 discusses the ndings along with their implications.

2 The game and research questions

The game For our experimental investigation, we introduce two versions of a standard linear public-good game implementing a voluntary contribution mechanism with n players, n ≥ 2, and multiple punishment stages: the basic game and the opinion game. Both games consist of an endogenous (but nite) number of stages. In the rst step, each player i receives an endowment of e > 0 monetary units and decides on her contribution x_i to the public good, with 0 ≤ x_i ≤ e. Each monetary unit invested in the public-good has a marginal rate of per-capita return α, with 1/n < α <1.

In the second step, each player is informed about the individual contribu-

tions to the public-good and the interim payo which equals ˆ

π_i =e−x_i+α

n

X

j=1

x_j. (1)

Furthermore, each player i announces whether and to which of the other players she wishes to assign punishment points. Punishment points pi→j

reduce the payo of playerj according to the details described below. Filing an announcement ai→j, ai→j ∈ {0,1}, incurs a cost of f_a>0 fori.⁶

In step three, the announcements are made public knowledge, and in our opinion condition, the players who are neither the punisher nor the target of an announcement ai→j, i.e., all players k s.t. k /∈ {i, j}, may voice their opinion about the announcement. Opinions only take on one of two values, consent or dissent, and do not have any formal consequences for player i's action space and payos. Notice that without the previous announcement ai→j, player i is not allowed to assign punishment points to j under either treatment condition. In the basic condition, players are informed about all announcements, but cannot express their consent or dissent.

After players have voiced their opinions (if applicable), all players are informed about the number and the identity numbers of supporters in the fourth step. In this step, each player i simultaneously decides on the (inte- ger) number of punishment pointspi→j she assigns at her private costc(pi→j), wherep_i→j ∈[0, p^max]. The punishment technology is such that each punish- ment point reduces the interim payo of the punished player by ten percent, and therefore, we have a natural limit for punishment points, p^max = 10.⁷ Therefore, the payo equals

π_i = ˆπ_i×maxn

0,(1−0.1X

j6=i

pj→i)o

−X

j6=i

c(pi→j)−F_a, (2) whereF_adenotes the total number of announcements made byitimesf_aand the cost function c : {0,1,2, ...,10} 7→ IR is a strictly-monotone increasing function with c(0) = 0. All players are informed about the resulting payos.

If there has been at least one announcement to assign punishment points in step two, additional stages of steps 2 to 4 follow: we allow all players to make new announcements (each incurring costs of f_a). To avoid potential demand eects in the experiment, we do not impose a restriction of pun- ishment opportunities to those who have been punished in the prior stage

6This procedure is designed to keep experimental subjects from announcing punishment actions just in case against every other subject.

7We adopt the punishment mechanism already used by Fehr and Gächter (2000) and Nikiforakis (2008).

as, e.g., in the design of Nikiforakis (2008). Again, in the opinion condi- tion, players not directly aected by an announcement of player i against j simultaneously voice their opinion on the new announcements. New an- nouncements allow players to increase the number of punishment points, even for players who have not been punished before.⁸ At the time of making their punishment-related decisions, players are provided with information about the accumulated points assigned to themselves and about their origin, the accumulated points received by other group members and the resulting pay- os, alongside the initial contributions to the public good made by each of the players. Thus in every iteration, information is provided that is may provide a basis for norm-guided or retaliative punishment. We repeatedly allow for new announcements and increases in punishment points until no player makes a further announcement to punish.⁹ Notice that players can only apply for and execute further punishment if this does not cause their own current payo π_i to become negative. Therefore, the number of iter- ations is nite and restricted at the most to P

iπˆ_i/f_a. Finally, players are informed about the payos and the game ends.

Predictions Since subjects play the game repeatedly over a nite number of rounds with changing anonymous interaction partners, the equilibrium of the game in both treatment conditions is rather obvious according to stan- dard theory in which any player will only be concerned with his own mon- etary payo. On the equilibrium path of the unique subgame-perfect Nash equilibrium, nothing changes compared to the standard public-good game.

If a player deviates making an announcement, other players are indier- ent between endorsing and dissenting from the announced action. Whether it is endorsed or not, the player making the announcement does not have any incentive to carry out the punishment, as this is costly to her. An- ticipating this, no player will contribute to the public-good, since it is by

∂ˆπ_i/∂x_i =−1 +α <0a dominant strategy not to do so.

Thus, one can interpret contributions as voluntary cooperation rates. In experiments, players often cooperate. Without developing a theoretic model of positive reciprocity here (see, e.g., Falk & Fischbacher, 2006), in light of the broad experimental evidence on voluntary public-good games (e.g., Isaac et al., 1985, or the recent surveys by Zelmer, 2003, or Gächter & Her- rmann, 2009), we expect players to contribute to the public-good. Further-

8Individual punishment costs are calculated according to the sum of points assigned per player, so that rationing the distribution of points across stages does not decrease costs.

9This procedure is similar to the one used by Nikiforakis and Engelmann (2009) in

more, as shown by Ostrom et al. (1992), Fehr and Gächter (2000), and many others, players are willing to sacrice own payo in order to punish others.

Research questions When thinking of social norms, a number of ques- tions arise that will be subsequently examined in this article. In the only study comparing dierent norm candidates for prosocial punishment, Car- penter and Matthews (2009) provide evidence in favor of absolute norms.

Notice, however, that this result is obtained in a setting where groups re- mained constant for the entire duration of the experiment. Thus, one can consider our framework as a robustness check for changing group composi- tions addressing the following question:

RQ 1. Do absolute contribution norms organize the decisions on whether to announce punishment, to agree to punishment, and how harshly to punish a player better than relative contribution norms?

Our second research question is concerned with the nature of the norm:

does it act only in one direction, explaining punishment of those who un- derprovide with respect to the norm, or does it also explain punishment of those who deviate positively from the norm? By examining this question, we are able to learn something about the motivation for antisocial punish- ment. In a post-experimental questionnaire, Fehr and Gächter (2000) asked subjects about the reasons for punishing high-contributors. The answers fall into ve categories: (i) random errors; (ii) the contribution level of the high-contributor is still not high enough; (iii) to increase one's relative payo advantage; (iv) anticipatory revenge against those who might sanction the antisocially punishing player in the current round; and (v) revenge against those who might have sanctioned the player in the previous round (even though, in Fehr and Gächter's case, these could not be identied). In our de- sign, while not impossible, random errors are rather unlikely, as players have to make two random mistakes in a row to exert unwanted punishment: they can always assign 0 points after an announcement.¹⁰ Category (ii) would simply mean that the norm is mis-specied. If this was indeed the case, it would show up in our absolute-norm model as a high absolute norm. Finally, categories (iii)-(v) concern the distinction between point assignments out of revenge, or retaliation, and antisocial punishment not triggered by received punishment points, be it out of spite or competitive thinking. By means of

10Such errors are rare: in basic, the fraction of 0-choices after an announcement is 3%, while it is 16% in opinion; in the latter, however, the number is largely driven by occasions in which neither player allowed to voice her opinion favored punishment.

our design, we are able to address this distinction. Therefore, to recapitulate, our second research question is

RQ 2. Does antisocial punishment as opposed to retaliation (i.e., punish- ment triggered by received punishment points) signicantly contribute to explaining decisions on whether to announce punishment and to punish a player? Are there dierences over punishment stages?

Finally, let us discuss the new aspect of our experiment, the elicitation of bystanders' norms of cooperation applied in evaluating others' punishment actions. As described above, we opt to disclose these evaluations publicly, so as not to render them meaningless in the eyes of our subjects. How- ever, the public announcement of others' (dis)agreement may change behav- ior. Masclet et al. (2003) report a positive eect of (nonmonetary) social (dis)approval on cooperation in public-good games.¹¹ One reading of this result is that public social assessment of behavior leads to an increase in the degree to which players identify with their group, which in turn may foster cooperation. However, this eect should be much less pronounced if present at all as (i) in our setting, players' voiced (dis-)approval was a routinely elicited information rather than an intentional and directed message as in Masclet et al., and (ii) Noussair and Tucker (2007) have shown the eect of social approval to rapidly diminish over the course of the experiment. Hence, whether the display of information on others' evaluations of one's punishment endeavors has any direct eect on contribution behavior is rather doubtful, while it may inuence the level of point assignments. Nonetheless, we ex- pect this eect to be rather weak. A more interesting question in terms of our main topic is whether players employ dierent norms when they are in the role of the punisher than when they only act as `impartial observers'.

We therefore set out to answer our nal research question, focusing on the relationship between player roles and cooperation norms:

RQ 3. Does the norm for social approval dier from the norms for both announcements and punishment?

11Rege and Telle (2004) come to the same conclusion after conducting a treatment in which they remove players' anonymity altogether. There are interesting variations of public-good games with voting on (non-)enforced absolute cooperation norms (e.g., Walker et al., 2000, Margreiter et al., 2004, Kroll et al., 2007) and voting on providing or refunding the public-good (Fischer & Nicklisch, 2007).

Table 1: Individual punishment costs

pi→j 0 1 2 3 4 5 6 7 8 9 10

c(pi→j) 0 1 2 4 6 9 12 16 20 25 30

3 Experimental design

We parameterized our model as follows: let there be n = 4 players each endowed with e = 20 experimental currency units. We choose α = 0.4 and announcement costs equal f_a = 1. Finally, for the individual punishment costs, we adopt the cost function used in Fehr and Gächter (2000) and Niki- forakis (2008). The costs for player i punishing player j are given by the convex sequence for increasing pi→j shown in Table 1.

For recruitment, we used the software package ORSEE (Greiner, 2004), the experimental software was written using z-tree (Fischbacher, 2007); ex- periments were run at the University of Bonn Experimental Economics Lab- oratory (BonnEconLab). On the day, subjects were welcomed and asked to draw lots, in order to assign each of them to a cabin. They were asked to move to their cubicle straight away. Once all subjects were seated, the in- structions were handed to them in written form before being read aloud by the experimenter.¹² Subjects were given the opportunity to ask any questions concerning the game privately. After questions had been answered individ- ually, subjects were handed a questionnaire to test their understanding of the rules.¹³ Questionnaires were corrected individually, while wrong answers were explained privately.

Subjects played ten repetitions (rounds) of the game. To prevent the pos- sibility of forming an individual reputation, every player received an identi- cation number between 1 and 4 at the beginning of each repetition, which she retained for the duration of the round, but which changed randomly in the next one. Furthermore, in order to prevent the emergence of group-specic cooperation norms and to test whether there is a global norm for contribu- tions to the public-good, we randomly formed groups anew at the beginning

12At the beginning of the experiment, subjects were informed that an unspecied and unrelated second part would follow the public good experiment. This second part consisted of an unincentivized questionnaire concerning socio-demographic background information of participants.

13For a translated version of the instructions and the questionnaire, see Appendices A and B.

of each round out of a pool of 12 subjects (`stranger matching'), while the group composition remained constant within each round.

Altogether, 144 subjects, mostly students majoring in various elds par- ticipated in the experiment. Mean age was 24.3 years (standard deviation 6.7 years), 43 percent were females. Each subject participated only once in the experiment. Overall, our data set consists of twelve independent groups of twelve subjects each yielding six independent observations for each treatment condition. Subjects were paid according to the sum of accumulated payos gained within the ten repetitions. The experimental currency was converted into euros (at a rate of 25 units per euro) and subjects were paid individually to ensure players' anonymity. Each session lasted for approximately 120 min- utes, subjects earned on average 18.20 euros (standard deviation 9.16 euros, including a 4-euro show-up fee).

4 Results

4.1 Data overview

In Figure 1, we depict round-wise payos, contributions, and punishment aggregated over all matching groups for each treatment. Even though contri- butions start out slightly higher in opinion (12.9vs10.1; contribution levels in the rst, second, and third round are dierent at a level of p = 0.0782, p = 0.1093, and p = 0.1495, respectively), this dierence wears away very quickly. In line with the ndings of Noussair and Tucker (2007), we do not nd any dierence in later rounds, nor in the overall contribution level.¹⁴ In the nal round, we observe average contributions of half the endowment in both treatments. Furthermore, we do not nd any signicant dierences for aggregate punishment or eciency levels as measured by average payos.

In both treatments, average payos start just above the Nash-equilibrium benchmark of20experimental currency units and oscillate around a value of 24.5units towards the end. Average punishment points assigned fall from1.2 in the rst round to approximately 0.3 in the nal two for both treatments.

The average number of punishment iterations is only insignicantly higher in opinion (1.92 vs 1.72in basic, p= 0.8095).¹⁵

Looking at the decision of whether to punish or not, we nd that overall,

14The corresponding values arep= 0.2002for the fth round, p >0.4for all remaining rounds, andp= 0.6991, for the overall contribution level. Unless otherwise indicated, all (within-)treatment comparisons are done by two-tailed Mann-Whitney U-tests (Wilcoxon signed-rank tests) on the basis of matching-group averages.

15This dierence is reversed for medians, with medians of2in basic vs 1in opinion.

Figure 1: Average payos, contributions (both: left axis), and punishment (right axis) over time.

about 6% of all possible announcements are made (5.7% in basic, 6.2% in opinion). The time trend mirrors that of punishment in general: whereas in the rst round, 8.7% (7.8%) in basic (opinion) of the potential announce- ments are made, the corresponding gures for the nal round are 3.7% for both treatments. Again, the reported treatment dierences are far from being signicant.¹⁶ On the iterations dimension, we nd the highest announcement rate in the rst punishment stage (7.2%), followed by the third and second iterations with 5.3% and 4.1%, respectively.¹⁷

Before we proceed to estimate the norms guiding our subjects' punish- ment behavior, let us take a closer look at the general punishment patterns in the two treatments. For this purpose, we classify punishment actions according to the punishing and punished players' contribution ranks.

16The corresponding p-values arep= 0.9372,p= 0.6291, andp= 0.6171, for the overall announcement level and the rst- and nal-round levels, respectively.

17In the fourth iteration, we observe a rate of 4.3%, and for the pooled remaining iterations, the gure is5.1%.

4.2 Punishment patterns

When aggregating the data over all iterations, we note that there is no general treatment dierence with respect to the ranks of punishers and punished players; this applies to both announcements and punishment received. To describe punishment behavior in greater detail, we disaggregate the data by iterations. Notice that the number of instances of ongoing iterations beyond the third decreases rapidly, so that in order to rely on a sucient number of observations, we have to restrict our analysis to the rst three iterations of each round. We nd that the frequency of announcements is the same across treatments in iterations 1 and 2, but this frequency has a tendency to be higher in opinion in iteration 3 (p= 0.0910). To analyze punishment patterns further, we test which contribution ranks mete out punishment, and who receives the punishment points. To this end, contributions within the group of four players are ranked: the player with the highest contribution is denoted by max", the second-highest by 3", and so on.¹⁸ For this exercise, we abstract from the number of points assigned but only count punishment actions. We will elaborate more on the number of points assigned in section 4.3 when discussing the estimated norms.

For a rst rough picture of the emerging punishment patterns, we pro- vide Table 2. In this table, we show the frequency of punishment actions by iteration, treatment, and contributor rank, relative to the corresponding punishment opportunities.¹⁹ Looking at the rank of players who are subject to punishment (that is, comparing columns), there is no signicant treat- ment dierence in any of the iterations. On a more general level, by looking at each iteration's lower-left-hand corners in the table the impression may arise that there is more punishment of players with higher contribution ranks by players with lower ranks in opinion; however, this dierence is clearly insignicant (p= 0.2971).

In the following, we will take a closer look at individual iterations sepa- rately. In iteration 1, the maximum-contributor punishes signicantly more than other players without there being a treatment dierence. There is a signicant dierence (p = 0.0210), however, with respect to the minimum- contributor. In basic, virtually no minimum-contributor ever carries out a punishment action in the rst iteration, while in opinion, this is roughly as likely as punishment by a player ranking second or third in terms of contri-

18In case of a tie, contributors are assigned the higher rank (i.e., if there are two players who contributed the second-highest contribution, they both are grouped to 3").

19Note that the data provided in Table 2 is an aggregation of all data points, irrespective of their (in-)dependence. Of course, the signicance tests following below are conducted based on independent observations.

Table 2: Punishment actions by contribution ranks, as fractions of opportu- nities

punishment in BASIC punishment in OPINION Iteration 1

max 3 2 min overall max 3 2 min overall

max 0.00 0.02 0.10 0.32 0.12 0.03 0.01 0.05 0.24 0.09 max 3 0.00 0.01 0.07 0.28 0.09 0.01 0.00 0.03 0.16 0.05 3 2 0.01 0.00 0.00 0.24 0.05 0.01 0.01 0.00 0.11 0.03 2 min 0.00 0.00 0.01 - 0.00 0.04 0.03 0.04 - 0.04 min Iteration 2

max 3 2 min overall max 3 2 min overall

max 0.01 0.01 0.03 0.06 0.03 0.05 0.03 0.02 0.07 0.05 max 3 0.03 0.00 0.02 0.05 0.03 0.02 0.06 0.04 0.03 0.03 3 2 0.01 0.02 0.04 0.04 0.02 0.02 0.02 0.09 0.06 0.03 2 min 0.04 0.05 0.04 - 0.05 0.07 0.05 0.05 - 0.06 min Iteration 3

max 3 2 min overall max 3 2 min overall

max 0.00 0.04 0.00 0.03 0.02 0.08 0.04 0.03 0.09 0.06 max 3 0.00 0.00 0.00 0.10 0.03 0.04 0.13 0.06 0.14 0.09 3 2 0.00 0.00 0.00 0.04 0.01 0.06 0.06 0.00 0.05 0.05 2 min 0.03 0.00 0.04 - 0.02 0.09 0.00 0.09 - 0.07 min

Note: to be read as punishment from row-contributor to column-contributor.

butions.

In iteration 2, this dierence between treatments diminishes since punish- ment activities of minimum-contributors in basic increase. Overall, there are no dierences in punishment actions across treatments for any of the ranks (p > 0.6, all pair-wise comparisons), nor is there a dierence in punishment between ranks within either treatment.

Interestingly, in the third iteration, the dierence between basic and opinion in terms of punishment activities by minimum-contributors reap- pears, although the dierence between treatments is only weakly signicant (p= 0.0553). So, while there is no general tendency for higher-contributing players to be punished more frequently by lower-contributing players in opin-

ion as pointed out above, there seems to be a specic treatment dierence concerning minimum-contributors. Given we do not have conclusive evidence on what may motivate this dierence, we relegate its discussion to section 5.

The number of independent observations diminishes rapidly for most of the cells in iteration 3, so that little can be said due to a lack of data.

However, there is an interesting point that comes to mind when eye-balling Table 2: the positive frequencies in the upper left-hand corner of the third- iteration tables could be a sign of sanction enforcement, in the sense of a player punishing another for not punishing a non-cooperative third (as, e.g., suggested by Henrich & Boyd, 2001). However, the actions represented by these fractions are too few and can partially also be attributed to other potential explanations like retarded punishment actions. As a consequence, it is impossible to pin-point most of these actions as sanction enforcement.

4.3 Contribution norms

4.3.1 Econometric models

To identify the determinants of players' behavior in our public-good game, we will compare the inuence of two relative and 21 absolute norms for all three punishment-related decisions of our experiment: the decision to announce punishment, the `opinion decision', and the actual punishment decision. For each iteration, we will estimate coecients and absolute norms separately, so that we can identify whether the estimated cooperation norms are stable across iterations. As mentioned before, the number of instances of ongoing iterations beyond the third decreases rapidly. In order to rely on a sucient number of observations, we restrict our analysis to the rst three iterations of each round.

For the analysis of announcements as well as of the opinions elicited we apply a probit regression with individual error clusters. Thus, we estimate the vector of coecients β for the basic econometric models

probit⁻¹(Prob(a^t,m_i→j = 1)) =x⁰β+ς_i+u_t,m , (3) and

probit⁻¹(Prob(v_k:i→j^t,m = 1)) =x⁰β+ς_k+u_t,m , (4) where Prob(a^t,m_i→j = 1) (Prob(v^t,m_k:i→j = 1)) stands for the latent probability that i announces to punishj in round t and iteration m (that k endorsesi's announcement to punish j in round t and iteration m), x for the matrix of regressors, ς_i for a vector of (unobserved) individual error clusters, and u_t,m

For the analysis of punishment decisions, we apply a tobit regression with individual error clusters. Thus, for the basic econometric model

ˆ

p^t,m_i→j =x⁰β+ς_i+u_t,m , and

p^t,m_i→j =







10 if pˆ^t,m_i→j >10, ˆ

p^t,m_i→j if 0<pˆ^t,m_i→j ≤10, 0 if pˆ^t,m_i→j ≤0,

(5)

we estimate the vectorβ, wherepˆ^t,m_i→j stands for the latent number of punish- ment points i assigns toj in round t and iteration m, and p^t,m_i→j is restricted to the interval [0,10].

In our quest to identify the norm governing punishment, we compare ve models each for the announcement decision, the voiced opinions, and the punishment decision. The rst model contains neither an absolute nor a relative norm, but only the control variables (see below), allowing us to assess the importance of either norm for punishment by comparison to the rst model. The second and third models test the importance of dierent relative norms, a group's average contribution and the punisher's own contribution, respectively. Models 4 and 5 test for an absolute norm.

Norm variables For models 2 to 4 (5), we dene two (one) distance mea- sures each. For each of these models, we measure the absolute dierences between the reference value under review and the contribution of the player to be punished, treating upward and downward deviations separately. The deviation terms are always dened by

n⁻ := |min{x^t_j −x˜^t,0}|, and

n⁺ := max{x^t_j−x˜^t,0}. (6)

where x˜^t is the respective reference value, and n⁻ (n⁺) denotes the corre- sponding downward (upward) deviation from this value. A summary of the models and their reference points is given in Table 3. Note that the variable n⁻ decreases in the punished player's contribution as long as this contribu- tion is below the respective reference point. A signicant positive eect of n⁻ would indicate that prosocial punishment is guided by the correspond- ing norm. If a norm determines antisocial punishment, we expect to nd a signicant positive eect of n⁺.

Notice that for all norms we face another potential estimation result in terms of the norm coecients: a positive coecient for r⁻_?,r⁻_??, ora⁻, imply- ing that negative norm violations increase (the probability of) punishment,

Table 3: Overview of the estimated norms norm terms denition

Model 1

Model 2 r⁻_? |min{x^t_j−x^t,0}|

r⁺_? max{x^t_j−x^t,0}

Model 3 r_??⁻ |min{x^t_j−x^t_i,0}|

r_??⁺ max{x^t_j−x^t_i,0}

Model 4 a⁻ |min{x^t_j −y,0}|

a⁺ max{x^t_j−y,0}

Model 5 a⁻ |min{x^t_j −20,0}|

Variables: x^t_j is the punished player's contribution; x^t,the average contribution;

x^t_i,the punisher's contribution;y, a constant integer number withy∈[0,20].

combined with a negative coecient for r_?⁺, r_??⁺, or a⁺, which would imply that a positive norm violation decreases the probability of punishment or the punishment level.²⁰ In this case, any deviation from contributing one's full endowment leads to an increase in the respective punishment determi- nant. In other words, subjects' elicited reference point would be nothing but their endowment, whereas the norm term merely identies the location of a kink on the right-hand side of the probit equation. Given the scenario just described is exactly what we observe, we add the absolute-norm model with y = 20 as a fth candidate to the models discussed. The dierence between the log-likelihoods of this model 5 and the best-performing model will give us a rst approximation of how much prediction power is lost by abstracting from the additional non-linearity. This can, of course, only be treated as a rough estimate in light of the fact that the full-contribution model by its very nature exhibits a lower number of free parameters.

In all models that include one of the norms detailed above, we allow that specic norm to act dierently in the two treatments. To incorporate this, we add an interaction eect between each norm part and a treatment dummy.

20Actually, there is yet another possibility, withr⁻_?,r⁻_??, anda⁻coecients being nega- tive, andr⁺_?,r_??⁺, ora⁺coecients being positive. This would mean that (the probability of) punishment increases in contributions, which, however, is rather counterintuitive and

Two nal remarks on our procedure seem warranted. The fourth model tests the importance of absolute norms. As in Carpenter and Matthews (2009), we do not allow the absolute norm to change over time in order to increase our ability to distinguish between the absolute and the relative norms. In our presentation of the results, we select and report that absolute norm tting the data best according to the log likelihood, based on a grid search over all possible contribution choices. This grid search is conducted for each deci- sion and each iteration separately, so that we allow absolute norms to dier.

However, assuming that there is an absolute standard guiding behavior, we should observe a consistent y over the dierent decisions and iterations.

Last but not least, notice that we retain the reference point of the punisher contribution in the regressions on voiced opinions, even though it is the bystander taking the decision, so that there could potentially be a change in the reference point. However, a model taking the bystander's contribution as a reference point (not reported here) is clearly outperformed by the reported model 3 on all iterations.

Controls Along with the inuence of relative and absolute norms, we con- trol for a number of other regressors that may inuence the decisions. For the analysis of the decisions on whether to announce punishment, and of how strongly to punish, those variables include the contribution of the player who punishes (x^t_i) and the sum of contributions of the two players not involved (X_k^t) from that particular round. We expect to nd positive eects for both as non-cooperators are typically prosocially punished by players who con- tribute a substantial amount to the public-good (see, e.g., Cinyabuguma et al., 2006), while free-riders may be more likely to be punished in coopera- tive groups for reasons of conformity. For potential temporal inuences (e.g., learning over the course of the experiment) we test by adding the variable round. Moreover, the dummy variable opinion marks those decisions from the opinion treatment. Additionally, for punishment decisions, we also in- clude the variablesum^t_v which counts the number of other players in favor of the punishment action in the opinion treatment, and which is zero for all observations from the basic treatment. Therefore, for punishment points, a negative (positive) eect of opinion indicates that there are less (more) points assigned in opinion than in basic if none of the players agrees with the punishment action in the former. However, a negative (positive) eect of sum^t_v indicates that in opinion, less (more) points are assigned if more of the others consent.

For the analysis of elicited opinions, we have to consider that all observa- tions come from the opinion treatment (thus, there is no treatment variable

in this regression), and that decisions are made by one of the `third par- ties'. Therefore, instead of the sum of contribution of the two players not involved, a regressor for the contribution of the player voicing her opinion (x^t_k) is included. Here, similar to the argument that players contributing larger amounts to the public-good are more likely to punish, we expect to nd a positive eect of the bystander's own contributions on the endorsement of punishment announcements.

Finally, for the regressions on decisions made in the second (third) iter- ation, we test for the potential eect of retaliation by means of the variable p^t,1_j→i (p^t,2_j→i) which measures the number of punishment points player i re- ceives from j in the rst (second) iteration. This variable in conjunction with the term for positive deviations from the norm allows us to answer our research question RQ 2: if punishment of high-contributors is guided by retaliation only, we should see signicant eects of p^t,m_j→i and no positive eect of a⁺,r_?⁺, or r⁺_??, respectively. If, however, there is antisocial behavior unrelated to revenge as a motive, the latter variables' coecients should be signicantly dierent from zero. For p^t,m_j→i we expect this to be the case, as according to the ndings of Nikiforakis (2008) and others, including a second punishment stage in a public-good game may trigger severe retaliation. In order to analyze dierences in retaliation across the two treatments, we in- clude the interaction eect p^t,1_j→i×opinion(p^t,2_j→i×opinion) in our regressions on announcements and on punishment points.

4.3.2 Estimation results

We organize our presentation of the results in the following way: rst, we discuss the ndings from our estimations on announcements and liken them to those on the assigned points. The discussion of potential treatment dier- ences is deferred to a second step. Finally, we present the estimations with respect to voiced opinions, to account for the treatment dierences in the level of point assignments.

In all regressions, an absolute term is included, which, however, is not reported. We compare between the nested models (model 1 versus model 2, 3, 4, and 5 respectively) on the basis of the Wald-chi²-test. Asterisks indicate signicance levels corresponding to this test.²¹ Other model comparisons are done on the basis of the test proposed by Vuong (1989). Unfortunately, for a majority of the comparisons, the test cannot be applied. In these

21∗∗∗ indicates signicance at a p <0.01level, ^∗∗ at ap <0.05level and^∗ at ap <0.1 level. Asterisks attached to log-likelihood values indicate the signicance level of the Wald-chi²-test comparing model 1 and the respective model.

Table 4: Mean marginal eects for announcements²¹

variable model 1 model 2 model 3 model 4 model 5 Iteration 1

x^t_i 0.0058^∗∗∗ 0.003^∗∗∗ −0.002^∗∗ 0.005^∗∗∗ 0.005^∗∗∗

X_k^t 0.0014^∗∗∗ −0.0003 0.002^∗∗∗ 0.002^∗∗∗ 0.002^∗∗∗

round −0.002^∗∗∗ 0.001^∗∗∗ −0.001^∗∗ −0.001^∗∗∗ −0.001^∗∗∗

opinion −0.012 0.012 0.003 0.007 0.038

r_?⁻/r⁻_??/a⁻ 0.017^∗∗∗ 0.012^∗∗∗ 0.01^∗∗∗ 0.01^∗∗∗

r_?⁺/r⁺_??/a⁺ −0.006 −0.005^∗∗ −0.009^∗

r_?⁻/r⁻_??/a⁻×op 0.007 −0.004 −0.005 −0.005 r_?⁺/r⁺_??/a⁺×op −0.005 0.005 0.011

best absolute norm 15 20

log likelihood −1027.5 −801.5^∗∗∗ −798.3^∗∗∗,a −809.4^∗∗∗ −813.5^∗∗∗

Iteration 2

x^t_i 0.001 0.0004 −0.0004 0.0008^∗∗ 0.0009^∗∗

X_k^t 0.0003 0.00004 0.0004 0.0005^∗ 0.0005^∗ round −0.001^∗∗∗ −0.0006^∗∗∗ −0.0006^∗∗∗ −0.0005^∗∗ −0.0005^∗∗∗

opinion −0.007 0.02 0.018 0.0012 0.033^∗

p^t,1_j→i 0.0158^∗∗∗ 0.018^∗∗∗ 0.017^∗∗∗ 0.018^∗∗∗ 0.018^∗∗∗

p^t,1_j→i×op 0.0013 −0.008 0.005 −0.013 0.0186 r_?⁻/r⁻_??/a⁻ 0.004^∗∗∗ 0.0027^∗∗∗ 0.0026^∗∗∗ 0.002^∗∗∗

r_?⁺/r⁺_??/a⁺ −0.001 −0.0008 −0.0022^∗

r_?⁻/r⁻_??/a⁻×op −0.003 −0.0014 −0.0006 −0.002 r_?⁺/r⁺_??/a⁺×op −0.002 −0.0005 0.004

best absolute norm 10 20

log likelihood −383.8 −370.7^∗∗∗ −369.34^∗∗∗ −363.7^∗∗∗,b −367.3^∗∗∗

Iteration 3

x^t_i 0.0001 0.0001 0.001 0.0001 0.0001

X_k^t 0.0007^∗∗ 0.0006^∗∗ 0.0006^∗∗ 0.0006^∗∗ 0.0006^∗∗

round 0.0003 0.0002 0.0003 0.0003^∗ 0.0003

opinion 0.0072 0.008 0.0076 0.0081 0.022^∗

p^t,2_j→i 0.012^∗∗ 0.0104^∗∗ 0.012^∗∗ 0.011^∗∗ 0.012^∗∗

p^t,2_j→i×op 0.0018 −0.0033 0.0021 0.0028 0.018 r_?⁻/r⁻_??/a⁻ 0.001^∗ 0.0003 0.0012^∗∗ 0.001^∗∗

r_?⁺/r⁺_??/a⁺ −0.0009 −0.0005 0.0008

r_?⁻/r⁻_??/a⁻×op −0.002 −0.0001 −0.0006 −0.0019 r_?⁺/r⁺_??/a⁺×op 0.0029 −0.0007 0.0067

best absolute norm 16 20

log likelihood −169.6 −166.3^∗∗ −168.6 −162.1^∗∗ −165.7^∗∗

Note: ^a (^b) model ts signicantly better than the second best model at p < .1

instances, we have to rely on a comparison of the log-likelihoods which, as a consequence, only provides a tentative answer of which model to prefer.

Norm estimations Results for the estimations of mean marginal eects on announcements are reported in Table 4, those for point assignments in Table 5. The most striking nding in terms of the focus of our paper is that in all iterations and (virtually) all models, our estimation results point to a full-contribution norm: on the one hand, the announcement probability as well as the amount of points assigned increase in downward deviations from the respective reference point as hypothesized, on the other, they decrease in upward deviations in all models in iterations 1 and 2 (often signicantly, particularly in the best-performing models). In iteration 3, there is a single announcement model for which the corresponding coecient is positive, even if insignicant (note that for point assignments, none of the reference-points contributes to explaining our data in this iteration). In other words, our esti- mation exercise de facto shows that the elicited reference point against which players' performance is measured is subjects' endowment in all iterations (but the third, for assignments). To summarize,

Result 1. The probability of an announcement is determined by the distance between the punished players' endowment and their contribution.²² Particu- larly, there is no reference value with the property that an increase in con- tributions above this value leads to an increase in the probability of being punished.

In other words, empirically there is no apparent norm (apart from the full- contribution benchmark) that distinguishes pro-social and anti-social or perverse punishment. If perverse punishment was norm-related behavior, there is no sign of it in our data.

The second main nding is that the application of the full-contribution standard diers between iterations. This can be seen from the fact that in iteration 1, model 3 performs best in all decision contexts (with a weakly signicant dierence to the next-best model for announcements), but that it is outperformed by absolute-norm models in subsequent iterations. For both announcements and point assignments (and opinions, but more on that later), behavior in the rst iteration is modulated strongly by the punisher's contribution. While the reference standard for who should be punished is (the punished) players' endowment as we have seen, the trigger for a punishment

22Research by Reuben and Riedl (2009) suggests that the determinant may be subjects' contribution capability rather than their endowment. Unfortunately, in our design the two cannot be discerned.

action often seems to be the potential punisher's contribution relative to that of the player to be punished. An intuitive explanation that has been proposed in the literature is that high-contributors do not want to be the sucker (e.g., Fehr & Gächter, 2000, Burlando & Guala, 2005). The larger the dierence between the two players' contributions, the stronger the emotional response (e.g., Fehr & Gächter, 2002, Xiao & Houser, 2005), and therefore, the more likely punishment is triggered. However, once the rst iteration is over, the importance of the punisher's relative contribution wears away. There may still be some second-order non-linearity with respect to the punished player's contribution level as indicated by the fact that the best t for models of type 4 is achieved for a y of 10 (iteration 2) and 16 (iteration 3, announcements;

for assignments,y= 17) but generally not much is to be gained by splitting the full-contribution norm of model 5.

Result 2. In the rst iteration, the announcement of punishment is triggered by the punisher's contribution relative to that of the punished player. This dierence in contributions also inuences strongly the level of punishment.

In later iterations, this is no longer the case.

Let us shortly review the eects of our control variables that, by and large, have the eects one might expect. The punisher's absolute contribution level has a positive eect on both announcements and point assignments, as do the contributions of the players who are neither the punisher nor the target of the punishment action;²³ the likelihood of an announcement decreases in the course of the experiment, as does the punishment level in iterations 2 and 3;

nally, the number of punishment points received in the preceding iterations is a strong indicator for both the likelihood and the level of punishment actions in iterations 2 and 3. Interestingly, in iteration 3, punishment points received have a negative impact on punishment assignments. A tentative explanation for this may be that, while subjects do not want to give in, they do start to economize on resources in this iteration, potentially in order not to nullify their round earnings completely.

Treatment eects The rst thing to note is that for announcements, none of the interaction variables across all models and iterations turns out to

23This holds true even for the third model, although the argument is a little more com- plex: in this model, we test for the inuence of the distance between the punisher's and the punishee's contribution. For that reason, the coecient for the punisher's contributionx^t_i measures the inuence of the level of both the punisher's and the punishee's contributions for a given distance. On the other hand, for a given punishee contribution, an increase in the announcing player's contribution leads to a higher distancer⁻_??, and thus, a higher probability of announcement, as stated above.

Table 5: Mean marginal eects for points²¹

variable model 1 model 2 model 3 model 4 model 5 Iteration 1

x^t_i 0.136^∗∗∗ 0.087^∗∗∗ −0.0707^∗ 0.196^∗∗∗ 0.196^∗∗∗

X_k^t 0.0445^∗∗∗ −0.006 0.0803^∗∗∗ 0.1^∗∗∗ 0.1^∗∗∗

round −0.057^∗∗ −0.029 −0.0287 −0.025 −0.028

opinion −2.76^∗∗∗ −0.465 −0.599 −5.59^∗∗∗ 1.43^∗

sum^t_v 5.036^∗∗∗ 4.324^∗∗∗ 4.071^∗∗∗ 4.27^∗∗∗ 4.27^∗∗∗

r_?⁻/r⁻_??/a⁻ 0.597^∗∗∗,b 0.428^∗∗∗,b 0.19 0.40^∗∗∗,b r_?⁺/r⁺_??/a⁺ −0.359 −0.241^∗∗ −0.44^∗∗∗,b

r_?⁻/r⁻_??/a⁻×op −0.599^∗∗∗,b −0.347^∗∗∗,b −0.174 −0.391^∗∗∗,b r_?⁺/r⁺_??/a⁺×op 0.415 0.253^∗∗ 0.431^∗∗∗,b

best absolute norm 3 20

log likelihood −1220.3 −1044.5^∗∗∗ −1040.7^∗∗∗ −1062.8^∗∗∗ −1063.0^∗∗∗

Iteration 2

x^t_i 0.084^∗ 0.054 −0.0232 0.118^∗∗ 0.123^∗∗

X_k^t 0.0342 0.0051 0.0525 0.071 0.070

round −0.082^∗∗ −0.077^∗∗ −0.072^∗ −0.0686^∗ −0.072^∗

opinion 0.416 2.132 1.761 −0.535 4.05^∗∗

sum^t_v 6.119^∗∗∗ 6.004^∗∗∗ 5.858^∗∗∗ 5.98^∗∗∗ 6.06^∗∗∗

p^t,1_j→i 1.777^∗∗∗ 2.286^∗∗∗ 2.422^∗∗∗ 2.379^∗∗∗ 2.379^∗∗∗

p^t,1_j→i×op −0.536 −1.05^∗ −1.399^∗∗ −1.11^∗ −1.15^∗ r_?⁻/r⁻_??/a⁻ 0.503^∗∗∗,b 0.357^∗∗∗,b 0.392^∗∗∗,b 0.35^∗∗∗,b r_?⁺/r⁺_??/a⁺ −0.158 −0.158 −0.287^∗

r_?⁻/r⁻_??/a⁻×op −0.482^∗∗,b −0.277^∗,b −0.106^b −0.355^∗∗,b r_?⁺/r⁺_??/a⁺×op −0.059 0.176 0.472^∗∗

best absolute norm 10 20

log likelihood −480.2 −468.1^∗∗∗ −465.5^∗∗∗ −462.5^∗∗∗ −465.2^∗∗∗

Iteration 3

x^t_i −0.0206 −0.014 0.02 −0.03 −0.03 X_k^t 0.077^∗∗∗ 0.0784^∗∗∗ 0.0654^∗∗∗ 0.069^∗∗∗ 0.069^∗∗∗

round 0.1^∗∗∗ 0.105^∗∗∗ 0.105^∗∗∗ 0.114^∗∗∗ 0.110^∗∗∗

opinion −2.14^∗∗ −2.057^∗∗ −2.191^∗∗ −2.334^∗ −1.328

sum^t_v 4.197^∗∗∗ 4.783^∗∗∗ 4.571^∗∗∗ 4.409^∗∗∗ 4.559^∗∗∗

p^t,2_j→i −3.705^∗∗∗ −3.587^∗∗∗ −3.569^∗∗∗ −3.501^∗∗∗ −3.53^∗∗∗

p^t,2_j→i×op 4.319^∗∗∗ 4.363^∗∗∗ 4.299^∗∗∗ 4.222^∗∗∗ 4.245^∗∗∗

r_?⁻/r⁻_??/a⁻ −0.0183^a −0.014^a 0.0085 0.0026^a r_?⁺/r⁺_??/a⁺ −0.0285 0.0081 0.0385

r_?⁻/r⁻_??/a⁻×op −0.346^∗,a −0.178^∗,a −0.1065 −0.176^∗,a r_?⁺/r⁺_??/a⁺×op 0.095 0.065 0.395

best absolute norm 17 20

log likelihood −343.6 −339.1 −339.9 −338.5 −339.1 Note: ^a (^b) the sum of the norm and the interaction between the norm and